This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is known today. Each module consists of five video lectures with embedded quizzes, followed by an electronically graded homework assignment. Additionally, modules 1, 3, and 5 also contain a peer assessment.
The homework assignments will require time to think through and practice the concepts discussed in the lectures. In fact, a significant amount of your learning will happen while completing the homework assignments. These assignments are not meant to be completed quickly; rather you'll need paper and pen with you to work through the questions. In total, we expect that the course will take 6-12 hours of work per module, depending on your background.

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Conformal Mappings

We’ll begin this module by studying inverse functions of analytic functions such as the complex logarithm (inverse of the exponential) and complex roots (inverses of power) functions. In order to possess a (local) inverse, an analytic function needs to have a non-zero derivative, and we’ll discover the powerful fact that at any such place an analytic function preserves angles between curves and is therefore a conformal mapping! We'll spend two lectures talking about very special conformal mappings, namely Möbius transformations; these are some of the most fundamental mappings in geometric analysis. We'll finish this module with the famous and stunning Riemann mapping theorem. This theorem allows us to study arbitrary simply connected sub-regions of the complex plane by transporting geometry and complex analysis from the unit disk to those domains via conformal mappings, the existence of which is guaranteed via the Riemann Mapping Theorem.